So how exactly would I run this test? Should it be done outdoors where lighting is better? If I printed off some targets and taped them to my garage door would that work for testing the lens with the bracketing app? How do I know if the lens performs well vs. poorly with this particular test?

You don't need any application IMHO to test your lens. What you need though is to proceed systematically.

First you probably want to check if your lens is optically correct.

You need some flat target with a pattern that will allow you to judge resolution and will fill the frame at at least 20x of the focal length. (32 -50cm suggested for 16mm up to 1-2m for 50mm). Any 100% static target that please your eye can do. (Newspapers taped to garage doors i.e.) Be sure not to test the lens on some moving targets - such as trees or water.

Try to place your target under even light, such as shade or overcast.

Put your lens on the sturdy tripod in self timer (if not in hurry 10s is better than 2s) mode or use remote shutter.

Switch off the OSS.

Switch to Manual Focus.

Software corrections makes this tests hardly relevant, but without'em it will be even harder for You to judge corners. So at least set the sharpness in JPEG to zero.

Use large magňification to acquire proper focus. If not sure, try to adjust it for more than 1 capture and select the sharpest image later. You might use peaking, but don't relay only on it.

Take several shots at several FL and apertures and check them carefully in the computer.

After this test, you should made a valuable conclusion about overall resolution behavior of your lens.

From there, you can proceed further depending on the results...

If you find one side softer than other (ot one corner) your lens might be decentered (or you didn't align your target well, or your sensor isn't properly alligned). If you have tripod with reversable middle column, reverse it and take another shot of your target with the camera upside-down. If not, turn it in portrait mode and take another shot of the target. In the computer check if the soft part moved with the image or not. Make the conclusion based on that.

If you got good resolution results with the above tests, you might have a problem with AF. Here you can use the bracketing application, but you can also simply switch between MF and AF and compare results. Be sure to make large enough contrast area for AF.

If you find your MF images to be sharper than AF ones, you should take your camera and lens to the service claiming AF problems.